A372921 - OEIS

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A372921

Triangle read by rows: T(n, k) = (Sum_{i=0..n-k} (-1)^i * binomial(n-k, i) * A007559(n-i)) * n! / ((n-k)! * A007559(k)) for 0 <= k <= n.

1, 0, 1, 3, 6, 2, 18, 63, 36, 6, 189, 828, 684, 216, 24, 2484, 13365, 14400, 6660, 1440, 120, 40095, 255474, 339390, 206280, 65880, 10800, 720, 766422, 5645619, 8915508, 6707610, 2827440, 687960, 90720, 5040, 16936857, 141626232, 259137144, 232306704, 121519440, 39130560, 7680960, 846720, 40320

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..44.

FORMULA

T(n, k) = (T(n-1, k-1) + 3 * T(n-1, k)) * n for 0 < k < n with initial values T(n, 0) = A033030(n) and T(n, n) = A000142(n).

E.g.f. of column k: exp(-t) / (1-3*t)^(1/3) * (t / (1-3*t))^k.

E.g.f.: exp(x*t / (1-3*t) - t) / (1-3*t)^(1/3).

EXAMPLE

Triangle T(n, k) starts:

n\k : 0 1 2 3 4 5 6 7

======================================================================

0 : 1

1 : 0 1

2 : 3 6 2

3 : 18 63 36 6

4 : 189 828 684 216 24

5 : 2484 13365 14400 6660 1440 120

6 : 40095 255474 339390 206280 65880 10800 720

7 : 766422 5645619 8915508 6707610 2827440 687960 90720 5040

etc.

MATHEMATICA

T[n_, k_]:=n!SeriesCoefficient[Exp[-t]/ (1-3*t)^(1/3) * (t / (1-3*t))^k, {t, 0, n}]; Table[T[n, k], {n, 0, 8}, {k, 0, n}]//Flatten (* Stefano Spezia, May 18 2024 *)

PROG

(PARI) T(n, k) = { sum(i=0, n-k, (-1)^i * binomial(n-k, i) * prod(j=1, n-i, 3*j-2)) * n! / ((n-k)! * prod(m=1, k, 3*m-2)) }

CROSSREFS

Cf. A007559, A033030 (column 0), A000142 (main diagonal).

Sequence in context: A090774 A147995 A351232 * A367028 A349863 A306189

Adjacent sequences: A372918 A372919 A372920 * A372922 A372923 A372924

KEYWORD

nonn,easy,tabl

AUTHOR

Werner Schulte, May 16 2024

STATUS

approved